Decomposition of numbers into weight × level + jump

The main goal of this page is to provide a new way to see certain numbers by their decomposition into weight × level + jump. This decomposition can be seen as a generalization of the sieve of Eratosthenes (which is the particular case of the sequence of natural numbers). Applied to prime numbers, this decomposition is used to obtain a new classification of primes.

Principles

Principle of decomposition: we choose the smallest weight such that in the Euclidean division of a number by its weight, the remainder is the jump (first difference, gap). The quotient will be the level.

Principles of classification: if the number is not decomposable, it is not classified. If the weight is greater than the level then the number is classified by level, if not then it is classified by weight.

Decomposition of natural numbers, the sieve of Eratosthenes

The weight is the smallest prime factor of and the level is the largest proper divisor of . The natural numbers classified by weight are the composites + 1 and the natural numbers classified by level are the primes + 1. The decomposition into weight × level + jump of natural numbers is thus a reformulation of the sieve of Eratosthenes.